# Fungrim entry: b65d19

$\operatorname{Im}\!\left(\operatorname{atan}\!\left(x + y i\right)\right) = \frac{1}{4} \log\!\left(\frac{{x}^{2} + {\left(1 + y\right)}^{2}}{{x}^{2} + {\left(1 - y\right)}^{2}}\right)$
Assumptions:$x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x + y i \notin \left\{-i, i\right\}$
TeX:
\operatorname{Im}\!\left(\operatorname{atan}\!\left(x + y i\right)\right) = \frac{1}{4} \log\!\left(\frac{{x}^{2} + {\left(1 + y\right)}^{2}}{{x}^{2} + {\left(1 - y\right)}^{2}}\right)

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x + y i \notin \left\{-i, i\right\}
Definitions:
Fungrim symbol Notation Short description
Im$\operatorname{Im}(z)$ Imaginary part
Atan$\operatorname{atan}(z)$ Inverse tangent
ConstI$i$ Imaginary unit
Log$\log(z)$ Natural logarithm
Pow${a}^{b}$ Power
RR$\mathbb{R}$ Real numbers
Source code for this entry:
Entry(ID("b65d19"),
Assumptions(And(Element(x, RR), Element(y, RR), NotElement(Add(x, Mul(y, ConstI)), Set(Neg(ConstI), ConstI)))))